# isomorphic automorphism groups

As I am sure some of you may have noticed I'm doing a series of exercises by Rotman and I am finding difficulties. Now I unbeaten into this problem

Give an example of an abelian and a non-abelian group with isomorphic automorphism groups.

Can you help me?

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Hint: this is surprisingly easy. – Jack Schmidt May 19 '11 at 16:41
May be u can think of groups $G$ which has $|\text{Aut}(G)|=p$ for some prime $p$.Then u are through – user9413 May 19 '11 at 16:51
@Chandru: that's not possible unless $p=2$, and even then, $G$ has to be abelian. – Chris Eagle May 19 '11 at 16:55
@Chandru: please stop leaving unhelpful comments. It is annoying to indicate that a comment is incorrect because it is not possible to downvote comments so I would strongly recommend that you stop doing this or I may have to start deleting them. – Qiaochu Yuan May 19 '11 at 16:57
Please don't yell. – Arturo Magidin May 19 '11 at 17:35

Try $A=\mathbb{F}_2^2$, and $G=S_3$.

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This is correct...but there's something about the question that unsettles me. I don't understand how the OP can be all out of ideas: to me, the obvious thing to try is just compute the automorphism groups of some small groups and see whether you get any coincidences. You learn a lot by doing this: e.g. it is probably more useful to know the automorphism groups of your $A$ and $G$ above than it is to know the answer to this specific question. If the OP didn't know to try this "technique", we should have told him/her (I think Jack Schmidt's comment was along these lines).... – Pete L. Clark May 19 '11 at 17:32
If the OP knows that this is what s/he should do but can't actually carry it out, than s/he has a much more basic problem with the material that we are just papering over by giving an answer. I am not scolding you by any means, but there are issues here to keep in mind: we should be teaching people how to fish rather than catching fish for them, if possible. – Pete L. Clark May 19 '11 at 17:34
@Pete L.Clark . You're right, I'm taking the first steps in group theory, and so I asked for help not complete resolution exercise. In this type of exercises I can use brute force, in the sense that I am able to find all automorphisms of a group and then subsequent comparisons, the fact is that to find an example it took me a long time, then I'm asking whether there is indeed a theorem, or more, which allows me to limit the search to the types of groups – user11116 May 20 '11 at 7:32